Considering the hypothetical distance between consecutive reals.

I have been pondering about a hypothetical distance between consecutive real numbers. It seems a bit of a paradox, though I expect it will be shown to be a consistent picture. I'll be using recently-learned terminology which hasn't completely set in mind yet, so please have patience with me :)

Firstly, let's define a hypothetical function S_r from R->R which in essence is the real analog of the successor function from N->N . I realize that such a function is inherently poorly defined since "succeeding" a real number x by some finite, non-zero number y will give z. But of course, between x and z will be an infinite number of real numbers to fill the interval of y-length. Never-the-less, being a hypothetical question, the terminology should serve for purposes of discussion.

Now, let's say we have an x and y in the set of reals satisfying |x-y|=0. From this, it is apparent to me that we can deduce that x=y. However, what I don't understand is why we can't also deduce that x=S_r(y), or that y=S_r(x). What I mean is that if we define y to be the real successor to x, and if the distance between x and y, |x-y|, were some non-zero quantity, then y couldn't be the real successor to x since the interval [x,y] would contain an infinite number of reals n that satisfy x<n<y. So the distance between x and its successor y cannot be greater than 0. In other words, it must equal zero, or |x-y|=0. But from this equation, we were also able to deduce that x=y.

Suppose there were a way to rigorously define the distance between consecutive reals. This distance could not be non-zero, so it would have to be based on some definition of zero itself, let's call it null for reference. Now, given a rigorously defined S_r, if x=S_r(y), then |x-y|=null=0. And if |x-y|=0, then x=y. But since x is also equal to S_r(y), then x=y=S_r(y), or in other words, y=S_r(y)?

I feel like I'm dressing up something more simple in unnecessary amounts of symbolism. I guess what I'm trying to say is if the distance two consecutive reals can't be non-zero, then it must be zero? And if so, then a real and its real successor must have the same quantity?

The ideas of "consecutive real numbers" and a "successor function for real numbers" don't make sense. For any two real numbers a and b, there is another real number in between them, for eaxmple (a+b)/2.

The ideas of "consecutive real numbers" and a "successor function for real numbers" don't make sense. For any two real numbers a and b, there is another real number in between them, for eaxmple (a+b)/2.

I mentioned this as well, the fact that such a distance between consecutive reals would have to be based on some definition of zero itself, provided a rigorous definition. Is there no basis to think of it hypothetically?

I mentioned this as well, the fact that such a distance between consecutive reals would have to be based on some definition of zero itself, provided a rigorous definition. Is there no basis to think of it hypothetically?

There is no such function, so asking about the properties is pointless (and certainly won't reveal anything about the actual real numbers). Given a real number, there is no other real that could reasonably be identified as its "successor".

I mentioned this as well, the fact that such a distance between consecutive reals would have to be based on some definition of zero itself, provided a rigorous definition. Is there no basis to think of it hypothetically?

No, there is no basis to think of hypothetically. A notion of "consecutive real numbers" is something that does not exists. So we can't talk about it
Furthermore, this forum is not the place to talk about hypothetical, non-existent notions.

Is there something you would like to ask or discuss that is not just merely hypothetical?

This is just a misunderstanding. When micromass said "Is there something you would like to ask or discuss that is not just merely hypothetical?" I took it as a joke thinking he was highlighting the fact that my posts have a tendency of going "off the rails" to the hypothetical. So I said Touché. And when he said "Furthermore, this forum is not the place to talk about hypothetical, non-existent notions", I agreed, and that's all, there's no subtext here.

And I do understand why there cannot be consecutive real numbers, assuming they exist produces an inconsistent result after all (that the a real and its consecutive real are the same value) as I questioned about in my OP. I was asking this because I'm trying to explore multiple trains of thought related to zero and its arithmetic properties, and this idea seemed relevant.